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arXiv:2004.11030 · Physical Review Letters

Nonequilibrium Dynamics and Weakly Broken Integrability

Joseph Durnin, M. J. Bhaseen, Benjamin Doyon

Published Sep 20, 2021• 98 citations
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Abstract

Motivated by dynamical experiments on cold atomic gases, we develop a quantum kinetic approach to weakly perturbed integrable models out of equilibrium. Using the exact matrix elements of the underlying integrable model, we establish an analytical approach to real-time dynamics. The method addresses a broad range of timescales, from the intermediate regime of prethermalization to late-time thermalization. Predictions are given for the time evolution of physical quantities, including effective temperatures and thermalization rates. The approach provides conceptual links between perturbed quantum many-body dynamics and classical Kolmogorov-Arnold-Moser theory. In particular, we identify a family of perturbations which do not cause thermalization in the weakly perturbed regime.

DOI
10.1103/physrevlett.127.130601

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Motivated by dynamical experiments on cold atomic gases, we develop a quantum kinetic approach to weakly perturbed integrable models out of equilibrium. Using the exact matrix elements of the underlying integrable model, we establish an analytical approach to real-time dynamics. The method addresses a broad range of timescales, from the intermediate regime of prethermalization to late-time thermalization. Predictions are given for the time evolution of physical quantities, including effective temperatures and thermalization rates. The approach provides conceptual links between perturbed quantum many-body dynamics and classical Kolmogorov-Arnold-Moser theory. In particular, we identify a family of perturbations which do not cause thermalization in the weakly perturbed regime.
There has been substantial progress in recent years in the quantitative understanding of the nonequilibrium time evolution of quantum fields. Important topical applications, in particular in high energy particle physics and cosmology, involve dynamics of quantum fields far away from the ground state or thermal equilibrium. In these cases, standard approaches based on small deviations from equilibrium, or on a sufficient homogeneity in time underlying kinetic descriptions, are not applicable. A particular challenge is to connect the far-from-equilibrium dynamics at early times with the approach to thermal equilibrium at late times. Understanding the ``link'' between the early- and the late-time behavior of quantum fields is crucial for a wide range of phenomena. For the first time questions such as the explosive particle production at the end of the inflationary universe, including the subsequent process of thermalization, can be addressed in quantum field theory from first principles. The progress in this field is based on efficient functional integral techniques, so-called n-particle irreducible effective actions, for which powerful nonperturbative approximation schemes are available. Here we give an introduction to these techniques and show how they can be applied in practice. Though we focus on particle physics and cosmology applications, we emphasize that these techniques can be equally applied to other nonequilibrium phenomena in complex many body systems.
There has been substantial progress in recent years in the quantitative understanding of the nonequilibrium time evolution of quantum fields. Important topical applications, in particular in high energy particle physics and cosmology, involve dynamics of quantum fields far away from the ground state or thermal equilibrium. In these cases, standard approaches based on small deviations from equilibrium, or on a sufficient homogeneity in time underlying kinetic descriptions, are not applicable. A particular challenge is to connect the far-from-equilibrium dynamics at early times with the approach to thermal equilibrium at late times. Understanding the ``link'' between the early- and the late-time behavior of quantum fields is crucial for a wide range of phenomena. For the first time questions such as the explosive particle production at the end of the inflationary universe, including the subsequent process of thermalization, can be addressed in quantum field theory from first principles. The progress in this field is based on efficient functional integral techniques, so-called n-particle irreducible effective actions, for which powerful nonperturbative approximation schemes are available. Here we give an introduction to these techniques and show how they can be applied in practice. Though we focus on particle physics and cosmology applications, we emphasize that these techniques can be equally applied to other nonequilibrium phenomena in complex many body systems.
We study the effects of integrability-breaking perturbations on the nonequilibrium evolution of many-particle quantum systems. We focus on a class of spinless fermion models with weak interactions. We employ equation of motion techniques that can be viewed as generalizations of quantum Boltzmann equations. We benchmark our method against time-dependent density matrix renormalization group computations and find it to be very accurate as long as interactions are weak. For small integrability breaking, we observe robust prethermalization plateaux for local observables on all accessible time scales. Increasing the strength of the integrability-breaking term induces a "drift" away from the prethermalization plateaux towards thermal behavior. We identify a time scale characterizing this crossover.
In this thesis we study three examples of interacting many-body systems undergoing a non equilibrium time evolution. Firstly we consider the time evolution in an integrable system: the sine-Gordon field theory in the repulsive regime. We will focus on the one point function of the semi-local vertex operator eiβφ(x)/2 on a specific class of initial states. By analytical means we show that the expectation value considered decays exponentially to zero at late times and we determine the decay time. The method employed is based on a form-factor expansion and uses the "Representative Eigenstate Approach" of Ref. [73] (a.k.a. "Quench Action"). In a second example we study the time evolution in models close to "special" integrable points characterised by hidden symmetries generating infinitely many local conservation laws that do not commute with one another, in addition to the infinite commuting family implied by integrability. We observe that both in the case where the perturbation breaks the integrability and when it breaks only the additional symmetries maintaining integrability, the local observables show a crossover behaviour from an initial to a final quasi stationary plateau. We investigate a weak coupling limit, identify a time window in which the effects of the perturbations become significant and solve the time evolution through a mean-field mapping. As an explicit example we study the XYZ spin-1/2 chain with additional perturbations that break integrability. Finally, we study the effects of integrability breaking perturbations on the non-equilibrium evolution of more general many-particle quantum systems, where the unperturbed integrable model is generic. We focus on a class of spinless fermion models with weak interactions. We employ equation of motion techniques that can be viewed as generalisations of quantum Boltzmann equations. We benchmark our method against time dependent density matrix renormalisation group computations and find it to be very accurate as long as interactions are weak. For small integrability breaking, we observe robust prethermalisation plateaux for local observables on all accessible time scales. Increasing the strength of the integrability breaking term induces a "drift" away from the prethermalisation plateaux towards thermal behaviour. We identify a time scale characterising this crossover.
We employ equation-of-motion techniques to study the nonequilibrium dynamics in a lattice model of weakly interacting spinless fermions. Our model provides a simple setting for analyzing the effects of weak integrability-breaking perturbations on the time evolution after a quantum quench. We establish the accuracy of the method by comparing results at short and intermediate times to time-dependent density matrix renormalization group computations. For sufficiently weak integrability-breaking interactions we always observe prethermalization plateaus, where local observables relax to nonthermal values at intermediate time scales. At later times a crossover towards thermal behavior sets in. We determine the associated time scale, which depends on the initial state, the band structure of the noninteracting theory, and the strength of the integrability-breaking perturbation. Our method allows us to analyze in some detail the spreading of correlations and in particular the structure of the associated light cones in our model. We find that the interior and exterior of the light cone are separated by an intermediate region, the temporal width of which appears to scale with a universal power law t1/3{t}^{1/3}t1/3.
After a sudden disruption, weakly interacting quantum systems first relax to a prethermalized state that can be described by perturbation theory and a generalized Gibbs ensemble. Using these properties of the prethermalized state we perturbatively derive a kinetic equation which becomes a quantum Boltzmann equation in the scaling limit of vanishing interaction. Applying this to interaction quenches in the fermionic Hubbard model we find that the momentum distribution relaxes to the thermal prediction of statistical mechanics. For not too large interaction, this two-stage scenario provides a quantitative understanding of the time evolution leading from the initial pure via a metastable prethermal to the final thermal state.